j=j(qFj,qF-j,qFiI)   j, .


The risk aversion level of generators is considered by using the CVaR measure. CVaR is a coherentA functional \(\mathcal {R}:\mathcal {L}^2\rightarrow (\infty ,\infty ]\)  is called a coherent measure of risk in the extended sense if (1) \(\mathcal {R}(C)=C\)  for all constants \(C\) ,
(2) \(\mathcal {R}((1-\lambda )X +\lambda X^{\prime }) \le (1-\lambda )\mathcal {R}(X)+\lambda \mathcal {R}(X)\)  for \(\lambda \in (0,1)\)  ("convexity"),
(3) \(\mathcal {R}(X)\le \mathcal {R}(X^{\prime })\)  ("monotonicity"),
(4) \(\mathcal {R}(X)\le 0\)  when \(||X^k-X||_2 \rightarrow 0\)  ("closedness"),
(5) \(\mathcal {R}(\lambda X)=\lambda \mathcal {R}(X)\text{for}\lambda >0\)  ("positive homogeneity").

risk measure that calculates VaR and optimizes CVaR simultaneously.
The CVaR at \(\alpha \)  confidence level can be defined as the expected value of the profit smaller than the (\(1-\alpha \) )-quantile of the profit distribution. In the scenario-based stochastic optimization method, CVaR\(_{\alpha }\)  measures the expected profit in the \((1-\alpha )\times 100\%\)  worst scenarios.
The \((1-\alpha )-\) quantile of the profit distribution is known as VaR, which is the largest value ensuring that the probability of obtaining a profit less than that value is lower than \(1-\alpha \) , \(\forall \alpha \in [0,1]\) .
Thus, the profit maximization problem solved by the risk (neutral and averse) generators is formulated in ((REF )), where \(\sigma _{k\omega }\)  is the probability assigned by generator \(k\)  to scenario \(\omega \) , and \(1-\alpha \)  represents the level of significance associated with the CVaR.
The risk averse problem solved by each generator is [1]}:


k, k, qkF  (1-)kk+[k-1 1-kk ]
